I had to use Propensity score matching for my study. I used the MatchIt function in R and I studied what it actually does. I understand that the propensity score is calculated using Logistic regression and it is the probability (p) of the treatment ( or log[p/(1 − p)] ) which is used to balance the Case and control group, conditional upon the covariates. But I have read papers saying the propensity score is the same as a distance measure.. why is PS a distance? Aren't they slightly different? Shouldn't a distance be just an absolute difference of two propensity scores instead? (one from the Case and one from the Control group?) Also when I ran the MatchIt function and viewed the result after matching, the column (or variable) name for propensity scores was 'distance'... Am I understanding it wrong?
One way to conceive of distance scores is as you have stated, which is a value that represents the distance between two individuals. Thus, each individual would have a distance score for all other individuals. In this sense, the difference between propensity scores is indeed a distance score. The Mahalanobis distance would also be a distance score.
MatchIt conceives of a distance score is the value for each individual that is used to compute the distance score as described previously. A precise way to describe this might be to call it a "position score," so that the difference between two position scores is a distance score. This is just an issue of terminology. Note that when you request Mahalanobis distance matching in
MatchIt, no distance value is produced in the output. Clearly they are relying on this latter use of distance scores.